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Practical homomorphic encryption over the integers for secure computation in the cloud

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Abstract

We present novel homomorphic encryption schemes for integer arithmetic, intended primarily for use in secure single-party computation in the cloud. These schemes are capable of securely computing arbitrary degree polynomials homomorphically. In practice, ciphertext size and running times limit the polynomial degree, but this appears sufficient for most practical applications. We present four schemes, with increasing levels of security, but increasing computational overhead. Two of the schemes provide strong security for high-entropy data. The remaining two schemes provide strong security regardless of this assumption. These four algorithms form the first two levels of a hierarchy of schemes, and we also present the general cases of each scheme. We further elaborate how a fully homomorphic system can be constructed from one of our general cases. In addition, we present a variant based upon Chinese Remainder Theorem secret sharing. We detail extensive evaluation of the first four algorithms of our hierarchy by computing low-degree polynomials. The timings of these computations are extremely favourable by comparison with even the best of existing methods and dramatically outperform many well-publicised schemes. The results clearly demonstrate the practical applicability of our schemes.

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Notes

  1. However, note that our “N” schemes below provide security against more malicious snooping.

  2. Paillier supports computation of linear functions with known coefficients homomorphically by repeated addition.

  3. The condition \(a_1, a_2, a_1-a_2\ne 0\), \((\bmod ~p\), \(\bmod ~q)\) fails with exponentially small probability \(3(1/p+1/q)\). Thus, \(a_1\) and \(a_2\) are indistinguishable in polynomial time from \(a_1,a_2\leftarrow _{\tiny {\$}}\,[0,pq)\).

  4. This work was aided by a Microsoft Azure for Research sponsorship [60].

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Acknowledgements

This work was supported in part by Engineering and Physical Sciences Research Council (EPSRC) research grants EP/I028099/1 and EP/M004953/1 and National Key Research and Development Program of China research Grant 2016YFB1000103. Experimentation conducted on Microsoft Azure was supported by a Microsoft Azure for Research sponsorship.

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  1. De Montfort University, Leicester, UK

    James Dyer

  2. School of Computing, University of Leeds, Leeds, UK

    Martin Dyer & Jie Xu

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This study was funded by Engineering and Physical Sciences Research Council (EP/I028099/1 & EP/M004953/1) and National Key Research and Development Program of China (2016YFB1000103). Usage of Microsoft’s Azure cloud was funded by a Microsoft Azure for Research sponsorship.

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A preliminary version of this paper [39] was presented at IMACC 2017.

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Dyer, J., Dyer, M. & Xu, J. Practical homomorphic encryption over the integers for secure computation in the cloud. Int. J. Inf. Secur. 18, 549–579 (2019). https://doi.org/10.1007/s10207-019-00427-0

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